Finite Group Databasebeta

← #194768

Subgroups — #194768

Order 512.

1278 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2581C211yesmaximal, characteristic
2#34187C211yesmaximal, characteristic
3#20734C212yesmaximal
4C2×#1461C211yesmaximal, characteristic
5#9022C214yesmaximal
6#9014C214yesmaximal
7#8026C212yesmaximal
8C22×C4×Q8C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10C2×#431C2212yes—
11#2581—24no—
12#2581C418yescomplemented
13C2×#400C2212yes—
14C2×#395C2212yes—
15#2356—24no—
16#2355—24no—
17#2313C2214yes—
18#2310—21no—
19C2×#391—21no—
20C2×#389—21no—
21C2×#387—22no—
22#2285C2212yes—
23#2283—22no—
24C2×#382C2211yescharacteristic
25#2274—21no—
26#1824C2214yes—
27#1803—24no—
28#1787C2214yes—
29C4×#103—24no—
30#1461—24no—
31C2×#324C2214yes—
32#1294C2218yes—
33C23×Q8D411yescharacteristic
34C23×Q8C2311yescharacteristic
35C2×C4×Q8—22no—
36C2×C4×Q8—22no—
37C2×C4×Q8—22no—
38C2×C4×Q8—24no—
39C2×C4×Q8—22no—
40C2×C4×Q8—22no—
41C2×C4×Q8—24no—
42C2×C4×Q8C2×C414yes—
43C2×C4×Q8C2×C414yes—
44C2×C4×Q8D414yes—
45C2×#117—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C42—21no—
53C22×C42D412yes—
54C22×C42C2311yescharacteristic
55#431—24no—
56#431—24no—
57C2×C4⋊C8—22no—
58C2×#103—28no—
59C2×C22⋊C8C2314yes—
60C2×C4×C8—22no—
61#400—24no—
62#400C2×C418yes—
63#395—24no—
64#395C2×C418yes—
65#391—28no—
66#391—28no—
67#389—28no—
68#387—216no—
69#387—216no—
70#384D414yes—
71#383—24no—
72#382—24no—
73#382—22no—
74#382—22no—
75#382C2×C414yes—
76#381—24no—
77#380—22no—
78#380C2314yes—
79C4×C4⋊C4—24no—
80C4×C4⋊C4—28no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4D414yes—
83C4×C22⋊C4—24no—
84#376—24no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95C2312yes—
90#340—28no—
91#324—28no—
92C22×Q8—22no—
93C22×Q8—22no—
94C22×Q8—24no—
95C22×Q8—24no—
96C22×Q8—21no—
97C22×Q8—22no—
98C22×Q8—22no—
99C22×Q8—44no—
100C22×Q8C22⋊C414yes—
101C22×Q8C22×C412yes—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4C2×D412yes—
106C23×C4C2×D412yes—
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2411yescharacteristic
110C22×C8—24no—
111C4×Q8—28no—
112C4×Q8—28no—
113C4×Q8—28no—
114C4×Q8—28no—
115C4×Q8—28no—
116C4×Q8—28no—
117C4×Q8—44no—
118C4×Q8—44no—
119#117—44no—
120#117—44no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—28no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—48no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4C22⋊C414yes—
139C2×C4⋊C4C22×C412yes—
140C2×C4⋊C4C22×C414yes—
141C2×C4⋊C4C2×D414yes—
142C2×C4⋊C4C2×D418yes—
143C2×C22⋊C4—24no—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42—24no—
147C2×C42—24no—
148C2×C42—24no—
149C2×C42—28no—
150C2×C42—42no—
151C2×C42—42no—
152C2×C42C22⋊C418yes—
153C2×C42C22×C412yes—
154C2×C42C2×D412yes—
155C2×C42C2×D412yes—
156C2×C42C2×D412yes—
157C2×C42#4114yes—
158C4⋊C8—44no—
159#103—416no—
160C22⋊C8—24no—
161C22⋊C8—24no—
162C4×C8—44no—
163#95—28no—
164#95—28no—
165#95—28no—
166#95—24no—
167#95—28no—
168#95—28no—
169#95—28no—
170#95—24no—
171#95—22no—
172#95—24no—
173#95C22×C414yes—
174C24C22×D411yescharacteristic
175C2×Q8—28no—
176C2×Q8—28no—
177C2×Q8—28no—
178C2×Q8—28no—
179C2×Q8—24no—
180C2×Q8—44no—
181C2×Q8—44no—
182C2×Q8—44no—
183C2×Q8—48no—
184C2×Q8—42no—
185C2×Q8—44no—
186C2×Q8—44no—
187C2×Q8—48no—
188C22×C4—24no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—22no—
198C22×C4—28no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—44no—
211C22×C4C2×C22⋊C414yes—
212C22×C4C2×C22⋊C414yes—
213C22×C4C2×C22⋊C412yes—
214C22×C4C2×C22⋊C412yes—
215C22×C4#12011yescharacteristic
216C22×C4#12011yescharacteristic
217C22×C4#12012yes—
218C22×C4#12312yes—
219C22×C4#12312yes—
220C22×C4#13514yes—
221C22×C4#13714yes—
222C22×C4C23×C411yescharacteristic
223C22×C4C22×D412yes—
224C22×C4C22×D411yescharacteristic
225C22×C4C2×#4112yes—
226C2×C8—48no—
227C2×C8—44no—
228C4⋊C4—28no—
229C4⋊C4—28no—
230C4⋊C4—28no—
231C4⋊C4—24no—
232C4⋊C4—48no—
233C4⋊C4—48no—
234C4⋊C4—416no—
235C4⋊C4—42no—
236C4⋊C4—44no—
237C4⋊C4—44no—
238C4⋊C4—48no—
239C4⋊C4—48no—
240C22⋊C4—44no—
241C22⋊C4—44no—
242C42—28no—
243C42—28no—
244C42—48no—
245C42—48no—
246C42—88no—
247C42C4×D414yes—
248C42C4×D414yes—
249C42#12114yes—
250C42#12114yes—
251C23—22no—
252C23—22no—
253C23C22×C22⋊C411yescharacteristic
254C23C2×#12011yescharacteristic
255C23C2×#12311yescharacteristic
256C23C2×#13512yes—
257C23C2×#13712yes—
258Q8—28no—
259Q8—48no—
260Q8—44no—
261Q8—416no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—48no—
288C2×C4—48no—
289C2×C4—48no—
290C2×C4—42no—
291C2×C4—82no—
292C2×C4—88no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4#38612yes—
296C2×C4#38612yes—
297C2×C4#38614yes—
298C2×C4#41614yes—
299C2×C4#41914yes—
300C2×C4#45214yes—
301C2×C4#45214yes—
302C2×C4#48518yes—
303C2×C4C2×C4×D412yes—
304C2×C4C2×#12112yes—
305C8—84no—
306C22—24no—
307C22—24no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22C2×#38611yescharacteristic
313C22C2×#41611yescharacteristic
314C22C2×#41911yescharacteristic
315C22C2×#45212yes—
316C22#302012yes—
317C22#302512yes—
318C22C2×#48512yes—
319C4—22no—
320C4—22no—
321C4—22no—
322C4—22no—
323C4—44no—
324C4—44no—
325C4—44no—
326C4—84no—
327C4—88no—
328C4—82no—
329C4#187318yes—
330C2—21no—
331C2—22no—
332C2—21no—
333C2#1821514yes—
334C2C2×#187311yescharacteristic
335C2#2095411yescharacteristic
336C2#2095911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
12. #2581C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#25812
#341872
#207342
C2×#14612
#90222
#90142
#80262